Theorem rexm 3320

Description: Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)

Assertion

Ref	Expression
rexm	|- ( E. x e. A ph -> E. x x e. A )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rexm

Step	Hyp	Ref	Expression
1		df-rex 2312	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpl 102	. . 3 |- ( ( x e. A /\ ph ) -> x e. A )
3	2	eximi 1491	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x x e. A )
4	1, 3	sylbi 114	1 |- ( E. x e. A ph -> E. x x e. A )

Syntax hints:   -> wi 4   /\ wa 97   E.wex 1381   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-rex 2312

This theorem is referenced by:  eusvobj2  5498